Chapter 18

Introduction to QualityStatistics for Business and Economics 6th Edition

Chapter GoalsAfter completing this chapter, you should be able to: Describe the importance of statistical quality control for process improvementDefine common and assignable causes of variationExplain process variability and the theory of control chartsConstruct and interpret control charts for the mean and standard deviationObtain and explain measures of process capabilityConstruct and interpret control charts for number of occurrences

The Importance of QualityPrimary focus is on process improvementData is needed to monitor the process and to insure the process is stable with minimum varianceMost variation in a process is due to the system, not the individualFocus on prevention of errors, not detectionIdentify and correct sources of variationHigher quality costs lessIncreased productivityincreased saleshigher profit

VariationA system is a number of components that are logically or physically linked to accomplish some purpose

A process is a set of activities operating on a system to transform inputs to outputs

From input to output, managers use statistical tools to monitor and improve the process

Goal is to reduce process variation

Sources of VariationCommon causes of variation also called random or uncontrollable causes of variation causes that are random in occurrence and are inherent in all processesmanagement, not the workers, are responsible for these causes Assignable causes of variationalso called special causes of variation the result of external sources outside the system these causes can and must be detected, and corrective action must be taken to remove them from the processfailing to do so will increase variation and lower quality

Process VariationVariation is natural; inherent in the world around usNo two products or service experiences are exactly the sameWith a fine enough gauge, all things can be seen to differTotal Process VariationCommon Cause VariationAssignable Cause Variation=+

Total Process VariationTotal Process VariationCommon Cause VariationAssignable Cause Variation=+PeopleMachinesMaterialsMethodsMeasurementEnvironmentVariation is often due to differences in:

Common Cause VariationCommon cause variation naturally occurring and expected the result of normal variation in materials, tools, machines, operators, and the environmentTotal Process VariationCommon Cause VariationAssignable Cause Variation=+

Special Cause VariationSpecial cause variationabnormal or unexpected variationhas an assignable causevariation beyond what is considered inherent to the process Total Process VariationCommon Cause VariationAssignable Cause Variation=+

Stable ProcessA process is stable (in-control) if all assignable causes are removedvariation results only from common causes

Control ChartsThe behavior of a process can be monitored over time Sampling and statistical analysis are usedControl charts are used to monitor variation in a measured value from a process

Control charts indicate when changes in data are due to assignable or common causes

OverviewProcess CapabilityTools for Quality ImprovementControl ChartsX-chart for the means-chart for the standard deviationP-chart for proportionsc-chart for number of occurrences

X-chart and s-chart Used for measured numeric data from a processStart with at least 20 subgroups of observed valuesSubgroups usually contain 3 to 6 observations eachFor the process to be in control, both the s-chart and the X-chart must be in control

PreliminariesConsider K samples of n observations eachData is collected over time from a measurable characteristic of the output of a production processThe sample means (denoted xi for i = 1, 2, . . ., K) can be graphed on an X-chart The average of these sample means is the overall mean of the sample observations

PreliminariesThe sample standard deviations (denoted si for i = 1, 2, . . . ,K) can be graphed on an s-chartThe average sample standard deviation is

The process standard deviation, , is the standard deviation of the population from which the samples were drawn, and it must be estimated from sample data(continued)

Example: SubgroupsSample measurements:

Subgroup measuresSubgroup numberIndividual measurements(subgroup size = 4)Mean, xStd. Dev., s1231512171716211591811152014.513.019.02.5173.1621.826Average subgroup mean = Average subgroup std. dev. = s

Estimate of Process Standard Deviation Based on sAn estimate of process standard deviation is

Where s is the average sample standard deviationc4 is a control chart factor which depends on the sample size, n Control chart factors are found in Table 18.1 or in Appendix 13If the population distribution is normal, this estimator is unbiased

Factors for Control ChartsSelected control chart factors (Table 18.1)

nc4A3B3B42.7892.6603.273.8861.9502.574.9211.6302.275.9401.4302.096.9521.290.031.977.9591.180.121.888.9651.100.181.829.9691.030.241.7610.9730.980.281.72

Control Charts and Control LimitsProcess AverageUCL = Process Average + 3 Standard Deviations LCL = Process Average 3 Standard DeviationsUCLLCL+3- 3timeA control chart is a time plot of the sequence of sample outcomesIncluded is a center line, an upper control limit (UCL) and a lower control limit (LCL)

Control Charts and Control LimitsThe 3-standard-deviation control limits are estimated for an X-chart as follows:(continued) Where the value of is given in Table 18.1 or in Appendix 13

X-ChartThe X-chart is a time plot of the sequence of sample meansThe center line is

The lower control limit is

The upper control limit is

X-Chart ExampleYou are the manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For seven days, you collect data on five deliveries per day. Is the process mean in control?

X-Chart Example: Subgroup DataThese are the xi values for the 7 subgroupsThese are the si values for the 7 subgroups

DaySubgroup SizeSubgroupMeanSubgroup Std. Dev.123456755555555.326.594.895.704.077.346.791.852.271.281.992.612.842.22

X-Chart Control Limits SolutionA3 = 1.43 is from Appendix 13

X-Chart Control Chart SolutionUCL = 8.889LCL = 2.737 024681234567MinutesDayx = 5.813__Conclusion: Process mean is in statistical control

s-ChartThe s-chart is a time plot of the sequence of sample standard deviations

The center line on the s-chart is

The lower control limit (for three-standard error limits) is

The upper control limit is

Where the control chart constants B3 and B4 are found in Table 18.1 or Appendix 13

s-Chart Control Limits SolutionB4 and B3 are found in Appendix 13

s-Chart Control Chart SolutionUCL = 4.4960241234567MinutesDayLCL = 0s = 2.151_Conclusion: Variation is in control

Control Chart BasicsProcess AverageUCL = Process Average + 3 Standard Deviations LCL = Process Average 3 Standard DeviationsUCLLCL+3- 3Common Cause Variation: range of expected variabilitySpecial Cause Variation: Range of unexpected variabilitytime

Process VariabilityProcess AverageUCL = Process Average + 3 Standard Deviations LCL = Process Average 3 Standard DeviationsUCLLCL3 99.7% of process values should be in this rangetimeSpecial Cause of Variation: A measurement this far from the process average is very unlikely if only expected variation is present

Using Control ChartsControl Charts are used to check for process controlH0: The process is in control i.e., variation is only due to common causesH1: The process is out of control i.e., assignable cause variation existsIf the process is found to be out of control, steps should be taken to find and eliminate the assignable causes of variation

In-control ProcessA process is said to be in control when the control chart does not indicate any out-of-control conditionContains only common causes of variationIf the common causes of variation is small, then control chart can be used to monitor the processIf the variation due to common causes is too large, you need to alter the process

Process In ControlProcess in control: points are randomly distributed around the center line and all points are within the control limitsUCLLCLtimeProcess Average

Process Not in ControlOut of control conditions:

One or more points outside control limits6 or more points in a row moving in the same direction either increasing or decreasing9 or more points in a row on the same side of the center line

Process Not in ControlOne or more points outside control limitsUCLLCLNine or more points in a row on one side of the center lineUCLLCLSix or more points moving in the same directionUCLLCLProcess AverageProcess AverageProcess Average

Out-of-control ProcessesWhen the control chart indicates an out-of-control condition (a point outside the control limits or exhibiting trend, for example)Contains both common causes of variation and assignable causes of variationThe assignable causes of variation must be identifiedIf detrimental to the quality, assignable causes of variation must be removedIf increases quality, assignable causes must be incorporated into the process design

Process CapabilityProcess capability is the ability of a process to consistently meet specified customer-driven requirementsSpecification limits are set by management (in response to customers expectations or process needs, for example)The upper tolerance limit (U) is the largest value that can be obtained and still conform to customers expectationsThe lower tolerance limit (L) is the smallest value that is still conforming

Capability IndicesA process capability index is an aggregate measure of a processs ability to meet specification limitsThe larger the value, the more capable a process is of meeting requirements

Measures of Process CapabilityProcess capability is judged by the extent to whichlies between the tolerance limits L and U

Cp Capability Index Appropriate when the sample data are centered between the tolerance limits, i.e.

The index is

A satisfactory value of this index is usually taken to be one that is at least 1.33 (i.e., the natural rate of tolerance of the process should be no more than 75% of (U L), the width of the range of acceptable values)

Measures of Process CapabilityCpk IndexUsed when the sample data are not centered between the tolerance limitsAllows for the fact that the process is operating closer to one tolerance limit than the otherThe Cpk index is

A satisfactory value is at least 1.33(continued)

Process CapabilityExampleYou are the manager of a 500-room hotel. You have instituted tolerance limits that luggage deliveries should be completed within ten minutes or less (U = 10, L = 0). For seven days, you collect data on five deliveries per day. You know from prior analysis that the process is in control. Is the process capable?

Process Capability:Hotel Data

Process Capability:Hotel Example SolutionThe capability index for the luggage delivery process is less than 1. The upper specification limit is less than 3 standard deviations above the mean.

p-ChartControl chart for proportionsIs an attribute chartShows proportion of defective or nonconforming itemsExample -- Computer chips: Count the number of defective chips and divide by total chips inspectedChip is either defective or not defectiveFinding a defective chip can be classified a success

p-ChartUsed with equal or unequal sample sizes (subgroups) over timeUnequal sizes should not differ by more than 25% from average sample sizesEasier to develop with equal sample sizesShould have large sample size so that the average number of nonconforming items per sample is at least five or six(continued)

Creating a p-ChartCalculate subgroup proportionsGraph subgroup proportionsCompute average of subgroup proportionsCompute the upper and lower control limitsAdd centerline and control limits to graph

p-Chart Example

Subgroup number, iSample sizeNumber of successesSample Proportion, pi123150150150151217.1000 .0800.1133Average sample proportions = p

Average of Sample ProportionsThe average of sample proportions = pwhere: pi = sample proportion for subgroup i K = number of subgroups of size nIf equal sample sizes:

Computing Control LimitsThe upper and lower control limits for a p-chart are

The standard deviation for the subgroup proportions isUCL = Average Proportion + 3 Standard Deviations LCL = Average Proportion 3 Standard Deviations

Computing Control LimitsThe upper and lower control limits for the p-chart are(continued)Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0

p-Chart ExampleYou are the manager of a 500-room hotel. You want to achieve the highest level of service. For seven days, you collect data on the readiness of 200 rooms. Is the process in control?

p Chart Example:Hotel Data# Not Day# RoomsReady Proportion1200160.080 2200 70.035 3200210.105 4200170.085 5200250.125 6200190.095 7200160.080

p Chart Control Limits Solution

p Chart Control Chart Solutionp = .0864UCL = .1460LCL = .02680.000.050.100.151234567PDayIndividual points are distributed around p without any pattern. Any improvement in the process must come from reduction of common-cause variation, which is the responsibility of management.__

c-ChartControl chart for number of defects per itemAlso a type of attribute chartShows total number of nonconforming items per unit examples: number of flaws per pane of glassnumber of errors per page of codeAssume that the size of each sampling unit remains constant

Mean and Standard Deviationfor a c-ChartThe sample mean number of occurrences isThe standard deviation for a c-chart iswhere: ci = number of successes per item K = number of items sampled

c-Chart Center and Control Limits The control limits for a c-chart areThe center line for a c-chart isThe number of occurrences can never be negative, so if the calculated lower control limit is negative, set LCL = 0

Process ControlDetermine process control for p-chars and c-charts using the same rules as for X and s-charts

Out of control conditions:One or more points outside control limitsSix or more points moving in the same directionNine or more points in a row on one side of the center line

c-Chart ExampleA weaving machine makes cloth in a standard width. Random samples of 10 meters of cloth are examined for flaws. Is the process in control?Sample number1234567Flaws found2130510

Constructing the c-ChartThe mean and standard deviation are:The control limits are:Note: LCL < 0 so set LCL = 0

The completed c-ChartThe process is in control. Individual points are distributed around the center line without any pattern. Any improvement in the process must come from reduction in common-cause variationUCL = 5.642LCL = 0Sample number1 2 3 4 5 6 7 c = 1.7146543210

Chapter SummaryReviewed the concept of statistical quality controlDiscussed the theory of control chartsCommon cause variation vs. special cause variationConstructed and interpreted X and s-chartsObtained and interpreted process capability measures Constructed and interpreted p-chartsConstructed and interpreted c-charts